Lindström theorems in graded model theory

Abstract

Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the Löwenheim–Skolem property, the Tarski union property, and the Robinson property, among others. As necessary technical restrictions, we assume that the models are valued on finite MTL-chains and the language has a constant for each truth-value.

Introduction

First-order predicate logic is the best known example of a formal language whose model theory had a great impact on 20th century mathematics, from non-standard analysis to abstract algebra. The celebrated characterization of classical first-order logic obtained by Per Lindström in the 60s (published as [23]; a nice accessible exposition can be found in [19]) is a landmark in contemporary logic. The introduction of a notion of “extended first-order logic”, that encompassed a great number of expressive extensions of first-order logic, allowed Lindström to establish, roughly, that there are no extensions of classical first-order logic that would also satisfy the compactness and Löwenheim–Skolem theorems, so this logic is maximal in terms of expressive power with respect to these properties (other similar characterizations soon followed). Expressive extensions of first-order logic are commonly called “abstract logics”,¹ giving rise to the field of abstract or soft model theory (cf. [5], [6]). In this field, one uses “only very general properties of the logic, properties that carry over to a large number of other logics” ([5], p. 225). Common examples of such properties are compactness or the Craig interpolation theorem. Abstract model theory is concerned with the study of such properties and their mutual interaction.

In the context of mathematical fuzzy logic (MFL) the possibility of abstract model-theoretic results was briefly considered by Petr Hájek in a technical report from 2002 [20]. Later, two Lindström-style results for the important cases of $[0,1]$-valued Łukasiewicz and rational Pavelka logics appeared in the literature [8], [9]. However, this work was not meant to present a general framework for other fuzzy logics, but rather ad hoc non-trivial results for particular systems.

Interestingly, Hájek had shown that the analogues of Lindström's first theorem fail for some of the main first-order fuzzy logics (BL∀, Π∀, and G∀) with their standard semantics (i.e. truth-values in the interval $[0,1]$). Furthermore, Hájek also established that the result cannot be obtained for any of BL∀, $\mathrm{Ł}\forall$, Π∀, or G∀ with their general semantics (the algebra of truth-values being allowed to vary among the elements of the variety corresponding to the logic in question). In fact, Hájek's argument holds for any fuzzy logic w.r.t. its general semantics if

• it satisfies the compactness and Löwenheim–Skolem theorems,

• it has the usual propositional connectives $\{\vee ,\wedge ,\mathrm{\&},\to ,\bar{1},\bar{0}\}$ and the quantifiers ∃ and ∀, and

• the compactness and Löwenheim–Skolem theorems remain true when adding the Baaz–Monteiro △ connective.

Perhaps discouraged by these initial negative results, the MFL community has not attempted again, to the best of our knowledge, to build a corresponding abstract model theory.

In this paper we would like to show that such a theory is actually a viable one, at least under certain technical conditions. In particular, we will give a general framework (as general as we can see) in which the Lindström theorems hold. This will lead to two main restrictions:

• our algebras of truth-values will be finite, and

• we will have a truth-constant for each element of the algebra (allowing for the possibility that such constants be definable or, more generally, that the language has the same expressive power, in the sense described below, as the version with constants).

Both conditions are necessary. We are led to (1) by a result from [20] and, given (1), also (2) can be quickly seen to be necessary. The framework ends up being the same recently used in [2] for some pure model-theoretic results and in [15] for results connected to fuzzy constraint satisfaction.

We follow closely the classical arguments from [5], adapting the ideas and techniques to the many-valued context. In [12], a translation into a two-sorted first-order language provides a way to interpret formulas in the languages studied in this paper as making statements not about many-valued structures but classical two-sorted structures. Even though we use this trick in Proposition 7, we should stress here that the applicability of this translation is limited. Indeed, the Lindström-style theorems in this paper are not immediate consequences of the classical ones. For one thing, since the translation of atomic formulas into the two-sorted languages are identity formulas, these languages have only one relation symbol, namely equality. They may vary on the function symbols they possess depending on the signature of the original language but they only require one relation symbol. The definition of an abstract logic, however, allows for arbitrary signatures. Moreover, observe that the classical Lindström theorems for two-sorted first-order logic would consider arbitrary two-sorted structures (no restrictions on the finiteness of the domain of one of the sorts, no algebraic structure imposed on said sort, etc.). Hence, the framework of the classical Lindström theorem is not ours and one cannot just apply it gratuitously; results have to be established independently. A similar reason was mentioned in [1] as to why the results there do not just follow from the classical Fraïssé theorem. In fact, the limitations of the translation are also pointed out in [12].

The paper is organized as follows: In §2 we review the basic notions of graded model theory, introduce the definition of an abstract graded logic, give some examples and present the key model-theoretic properties that will be involved in our characterizations (abstract completeness, the Löwenheim-Skolem property, compactness, the Karp property, and the Tarski union property among others). In §3 we show that the first-order logic based on an arbitrary (but fixed) finite MTL-chain has all the desired model-theoretic properties introduced in the previous section. In §4 we establish our six Lindström maximality results. Finally, in §5 we end with some concluding remarks.